James-Stein-style estimator when we place greater importance on some components The James-Stein estimator allows us to get a better overall estimate of a mean vector (length $\ge 3$) than we would be able to get by estimating the components independently. My intuition is that, under the hood, it is willing to sacrifice some of the components in order to tighten up estimates of the others, resulting in tighter estimates overall.
This sounds nice, but it makes the assumption that we would be willing to sacrifice any component for any other, that the cost of missing one bad is offset by tightening up another estimate. This need not be the case. Further, I do not see a way that such an estimator makes clear which components are sacrificed, so I would not even know which marginal estimates are the best.
What happens when we want to emphasize one component more than the others or want to know which components are sacrificed for which others?
 A: If one wishes to favour the estimation of one component of the mean vector, the loss function can penalise the attached error more for that component. Using a quadratic loss,
$$\text{L}(\theta,\delta)=(\delta-\theta)^\text{T}Q(\delta-\theta)$$
where $Q$ is a symmetric positive semi-definite matrix, there exist James-Stein estimators that dominate the MLE $\varphi_0(X)$ for that loss, provided $Q$ is of rank $3$ or more, as e.g.
$$\varphi(x)=\varphi_0(x)-h(\varphi_0(x)^\text{T}B\varphi_0(x))C\varphi_0(x)$$
where $B$ is a symmetric symmetric positive definite matrix and $C$ is a symmetric matrix such that $B$ and $C$ share an eigenbasis. See our 1989 paper for details (under some conditions on the function $h$, obviously).
A: Here's an aspect of the James-Stein estimator that I haven't seen in the literature, although it may be somewhere. The result is based on the use of the mean squared error (L2-norm) for measuring the quality of an estimator. What the L2-norm does is that basically every error is multiplied, or in other words weighted, with itself. When we consider the different components, this means that components with big error dominate the MSE, and bringing big errors in components down at the expense of increasing smaller errors in the components will improve the MSE. What the James-Stein estimator does is to shrink observations with large norms, which have large absolute values in one or more components. Such large values can occur for two reasons: (1) the true component means are indeed large in absolute value, or (2) the true components means are not large, but the observation is in the tail region of its normal. Case (2) will cause a large MSE for the standard estimator. James-Stein will do worse in case (1) but better in case (2). Because of the L2-norm, the improvement in case (2) will be more beneficial in terms of MSE than the damage that is done in case (1). If the number of components is large enough, both cases will likely happen (even in 3-d the probability for case (2) to happen is big enough to account for the issue (1)).
As far as I see it, the JS-estimator "sacrifices" quality of component estimation of which the true means are large in absolute value for the sake of improving the estimation of the components the true means of which are small in absolute value. The latter have more potential to ruin the MSE, at least if we know we're observing something with relatively large absolute value. One way of interpreting this is not to say that the JS-estimator is truly better than the standard estimator, but rather to say that this is a consequence of using L2-loss rather than, say, L1-loss, the latter not being dominated by the larger component errors, which in many situations may be preferable (however deriving general estimation theory based on L1-loss is more tedious).
